Improve docs site search panel discoverability and empty state

## Goal

The docs site search panel (Starlight's built-in Pagefind search) supports keyboard shortcuts, but they are not discoverable by users. Investigate how to surface them and improve the empty state.

Two areas to explore:

- **Shortcut discoverability** — show the supported keyboard shortcuts as helper text at the bottom of the search panel
- **Empty state** — present a useful empty state, e.g. a few suggested articles or example searches, instead of a blank panel

## Expected Outcome

- An assessment of what Starlight/Pagefind expose for customizing the search modal (component overrides, slots, theming)
- A recommendation on how to render shortcut helper text and what shortcuts to surface
- A recommendation on the empty-state content (suggested articles vs. example queries) and how to source it
- A follow-up task or set of tasks if implementation is warranted

## Related Issues

- Originated from #1754

> [!IMPORTANT]
> Internal only — this issue is maintained by the core team and is not accepting external contributions.

Start by reviewing Starlight's built-in Pagefind search customization options and the repository's existing docs search configuration. Document whether component overrides, slots, or theming can support shortcut helper text and an empty state, recommend the content and source for each, and record any warranted follow-up tasks.

Written by the indexing model from the issue text.